Six republicans and four democrats have applied for two open positions on a planning committee. since all the applicants are qualified to serve, the city council decides to pick the two new members randomly. what is the probability that both come from the same party?

Step-by-step explanation:

Probabilities

When we choose from two different sets to form a new set of n elements, we use the so-called hypergeometric distribution. We'll use an easier and more simple approach by the use of logic.

We have 6 republicans and 4 democrats applying for two positions. Let's call R to a republican member and D to a democrat member. There are three possibilities to choose two people from the two sets: DD, DR, RR. Both republicans, both democrats and one of each. We are asked to compute the probability of both being from the same party, i.e. the probability is

Let's compute P(DD). Both democrats come from the 4 members available and it can be done in different ways.

For P(RR) we proceed in a similar way to get different ways.

The total ways to select both from the same party is

The selection can be done from the whole set of candidates in different ways, so

The hypotenuse is exactly sqrt(149) units long

This is approximately equal to 12.20656

Work Shown:

The right triangle has sides a,b,c such that

a = 7

b = 10

c = unknown hypotenuse (aka longest side)

Let's use the pythagorean theorem to solve for side c

a^2 + b^2 = c^2

7^2 + 10^2 = c^2

49 + 100 = c^2

149 = c^2

c^2 = 149

c = sqrt(149) which is the exact answer

c = 12.20656 which is the approximate answer

The approximate answer is rounded to five decimal places